Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Harnack inequality for the linearized
parabolic Monge-Ampère equation


Author: Qingbo Huang
Journal: Trans. Amer. Math. Soc. 351 (1999), 2025-2054
MSC (1991): Primary 35K10; Secondary 35B45
DOI: https://doi.org/10.1090/S0002-9947-99-02142-X
Published electronically: January 27, 1999
MathSciNet review: 1467468
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we prove the Harnack inequality for nonnegative solutions of the linearized parabolic Monge-Ampère equation

\begin{displaymath}u_{t}-\text{tr}((D^{2}\phi (x))^{-1}D^{2}u)=0\end{displaymath}

on parabolic sections associated with $\phi (x)$, under the assumption that the Monge-Ampère measure generated by $\phi $ satisfies the doubling condition on sections and the uniform continuity condition with respect to Lebesgue measure. The theory established is invariant under the group $AT(n)\times AT(1)$, where $AT(n)$ denotes the group of all invertible affine transformations on ${\mathbf{R}}^{n}$.


References [Enhancements On Off] (What's this?)

  • [Ca1] L.A. Caffarelli, Some regularity properties of solutions of Monge-Ampère equation, Comm. Pure Appl. Math. XLIV (1991), 965-969. MR 92h:35088
  • [Ca2] -, Boundary regularity of maps with convex potentials, Comm. Pure Appl. Math. XLV (1992), 1141-1151. MR 93k:35054
  • [Ca-C] L. A. Caffarelli and X. Cabré, Fully nonlinear elliptic equations, AMS Colloquium Publications V. 43, AMS, Rhode Island, 1993. MR 96h:35046
  • [Ca-Gu1] L. A. Caffarelli and C. E. Gutiérrez, Real analysis related to the Monge-Ampère equation, Trans. Amer. Math. Soc. 348 (1996), 1075-1092. MR 96h:35047
  • [Ca-Gu2] -, Properties of the solutions of the linearized Monge-Ampère equation, Amer. J. of Math. 119 (1997), 423-465. MR 98e:35060
  • [G-T] D. Gilbarg & N. S. Trudinger, Elliptic partial differential equations of second order, 2nd edition, Springer-Verlag, 1983. MR 86c:35035
  • [Gu] C. E. Gutiérrez, On the Harnack inequality for viscosity solutions of non-divergence equations, preprint.
  • [Gu-H] C. E. Gutiérrez & Q. Huang, Geometric properties of the sections of solutions of the Monge-Ampère equation, Trans. AMS. to appear.
  • [K-S] N. V. Krylov & M. V. Safonov, Certain properties of solutions of parabolic equations with measurable coefficients, Izvestia Akad. Nauk. SSSR 44 (1980), 161-175. MR 83c:35059
  • [M] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure. Appl. Math. 17 (1964), 101-134. MR 28:2356
  • [S] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Math. Series #43, Princeton U. Press, Princeton, NJ, 1993. MR 95c:42002
  • [T] K. S. Tso, On an Alekandrov-Bakel'man type maximum principle for second-order parabolic equations, Comm. PDE 10 (1985), 543-553. MR 87f:35031
  • [W] L. Wang, On the regularity theory of fully nonlinear parabolic equations I, Comm. Pure Appl. Math. 45 (1992), 27-76. MR 92m:35126
  • [W-W] Rou-Huai Wang & Guang-Lie Wang, On existence, uniqueness and regularity of viscosity solutions for the first initial-boundary value problems to parabolic Monge-Ampère equation, Northeast. Math. J. 8 (1992), 417-446. MR 94d:35085

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 35K10, 35B45

Retrieve articles in all journals with MSC (1991): 35K10, 35B45


Additional Information

Qingbo Huang
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
Address at time of publication: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
Email: qhuang@math.utexas.edu

DOI: https://doi.org/10.1090/S0002-9947-99-02142-X
Keywords: Harnack inequality, affine invariant, linear parabolic Monge-Amp\`{e}re equation, section
Received by editor(s): December 15, 1996
Received by editor(s) in revised form: May 13, 1997
Published electronically: January 27, 1999
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society